Overview
- Group
- SmallGroup(1296,2909)
- Rank
- 3
- Schläfli Type
- {4,6}
- Vertices, edges, …
- 108, 324, 162
- Order of s0s1s2
- 12
- Order of s0s1s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Orientable
Quotients maximal quotients in bold
3-fold
6-fold
9-fold
18-fold
27-fold
54-fold
81-fold
108-fold
162-fold
Covers minimal covers in bold
None in this atlas.
Irregular Quotients of which this is a minimal cover
Click an entry to reveal its facets and vertex figures.
P/N, where N=<(s0*s2*s1)^2*s0*s1*s2*s1*s0*(s1*s2)^2> of order 2
81 facets
- 81 of {4}*8
54 vertex figures
- 54 of {6}*12
P/N, where N=<(s0*s2*s1)^2*s0*s1*s2*s1*s0*(s1*s2)^2*s1> of order 2
81 facets
- 81 of {4}*8
57 vertex figures
P/N, where N=<s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 3
54 facets
- 54 of {4}*8
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 3
54 facets
- 54 of {4}*8
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^3*s1*s2> of order 3
54 facets
- 54 of {4}*8
36 vertex figures
- 36 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 3
54 facets
- 54 of {4}*8
36 vertex figures
- 36 of {6}*12
P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*(s1*s2)^2> of order 4
45 facets
30 vertex figures
P/N, where N=<(s0*s1)^2, s1*(s2*s1*s0)^2*s1*s2*s1*s0*(s1*s2)^2> of order 4
45 facets
27 vertex figures
- 27 of {6}*12
P/N, where N=<(s0*s1)^2, (s0*s2*s1)^2*s0*s1*s2*s1*s0*s2> of order 6
36 facets
18 vertex figures
- 18 of {6}*12
P/N, where N=<s0*(s1*s2)^2*s1*s0*s2, s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 6
27 facets
- 27 of {4}*8
21 vertex figures
P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 9
18 facets
- 18 of {4}*8
12 vertex figures
- 12 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 9
18 facets
- 18 of {4}*8
16 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)(27,62)(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,74)(76,77)(79,80);; s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);; s2 := ( 1,42)( 2,40)( 3,41)( 4,39)( 5,37)( 6,38)( 7,45)( 8,43)( 9,44)(10,32)(11,33)(12,31)(13,29)(14,30)(15,28)(16,35)(17,36)(18,34)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54)(55,69)(56,67)(57,68)(58,66)(59,64)(60,65)(61,72)(62,70)(63,71)(73,76)(74,77)(75,78);; poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP)
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)(27,62)(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,74)(76,77)(79,80); s1 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77); s2 := Sym(81)!( 1,42)( 2,40)( 3,41)( 4,39)( 5,37)( 6,38)( 7,45)( 8,43)( 9,44)(10,32)(11,33)(12,31)(13,29)(14,30)(15,28)(16,35)(17,36)(18,34)(19,49)(20,50)(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54)(55,69)(56,67)(57,68)(58,66)(59,64)(60,65)(61,72)(62,70)(63,71)(73,76)(74,77)(75,78); poly := sub<Sym(81)|s0,s1,s2>;
Finitely Presented Group Representation (Magma)
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.